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Abstract:

Due to time-inconsistency or policymakers' turnover, economic
promises are not always fulfilled and plans are revised
periodically. This fact is not accounted for in the commitment or
the discretion approach. We consider two settings where the planner
occasionally defaults on past promises. In the first setting, a
default may occur in any period with a given probability. In the
second, we make the likelihood of default a function of endogenous
variables. We formulate these problems recursively, and provide
techniques that can be applied to a general class of models. Our
method can be used to analyze the plausibility and the importance
of commitment and characterize optimal policy in a more realistic
environment. We illustrate the method and results in a fiscal
policy application.

Keywords: Commitment, discretion, fiscal
policy

JEL classification: C61, C63, E61,
E62

1 Introduction

1.1 Motivation and Contribution

In a general class of macroeconomic models, households' behavior
depends on expectations of future variables. Characterizing optimal
policy in such circumstances is intricate. A planner influences
households' expectations through its actions, and households'
expectations influence the actions of the planner. Following the
seminal papers by Kydland and Prescott (1977) and
Barro and Gordon (1983a), the literature has taken two
different approaches to deal with this problem - commitment and
discretion.

Both the commitment and discretion approaches are to some extent
unrealistic. The commitment approach does not match the observation
that governments have defaulted on past promises. The discretion
approach rules out the possibility that the government achieves the
benefits of making and keeping a promise, even if there is an a
posteriori incentive to default. It seems more reasonable to
assume that institutions and planners sometimes fulfill their
promises and sometimes do not.

This paper characterizes optimal policy in two frameworks where
some promises are kept while others are not. We first consider a
setting where current promises will be fulfilled with a given
probability. This setting can easily be extended to one where
promises are only kept during a finite tenure. Lastly, we make the
likelihood of default a function of endogenous variables. There may
be several interpretations for the loose commitment settings
just described. A political economy interpretation is that
governments fulfill their own promises but it is possible that a
new government is elected and the previous government's promises
are not considered. Another interpretation is that a government
commits to future plans, but defaulting becomes inevitable if
particular events arise, such as wars or political instability. As
it is common in the discretion literature, we consider that a
default on past promises occurs whenever a reoptimization takes
place. For the purposes of this paper it is inconsequential whether
the reoptimization is undertaken by the same planner or by a newly
appointed one. Another interpretation of these settings is that
policymakers are often required to reevaluate policy. Hence, the
optimal current policy recommendation should already take into
account that policy will be revised in the future.

The contribution of this paper is in part methodological. We
considerably generalize and extend the work of Roberds (1987) and Schaumburg and Tambalotti (2007). The methods previously
available can be applied to linear quadratic models but can not
usually be applied to non-linear models. The large bulk of models
used in macro have non-linear features, such as non-linear utility
and production functions. One could argue that non-linear models
can be approximated by linear-quadratic methods. However, taking a
linear quadratic approximation to a non-linear model often requires
the timeless perspective assumption, which sharply contradicts the
loose commitment assumption. This issue significantly
reduces the set of models that one can solve in a loose
commitment setting with linear quadratic techniques. We provide
a methodology that can be applied to a large class of models, and
we prove that the solution of these problems is recursive. In
addition, we extend the loose commitment approach to models
with endogenous state variables. This opens the possibility for
each planner to set these state variables strategically and
influence future planners. This issue raises some interesting
questions and poses methodological challenges.

As an illustration of what can be learnt, we provide an
application to fiscal policy. When the probability of keeping
promises is decreased from 1 to , most variables
move substantially towards discretion. Hence, in our application a
full commitment framework seems unrealistic. We then discuss the
effects of a default on economic variables. The main effect of
reneging on past promises is an increase in the capital tax. Policy
instruments are also found to change relatively more than private
allocations during a default. We then discuss our default and
commitment cycles in the spirit of the political business cycle
literature. We characterize how the welfare gains change as a
function of the probability to commit or the implied average time
period before a default. In the endogenous probability model, we
find that since the probability of commitment/reelection depends on
endogenous state variables, the planner actively manipulates these
state variables in order to enhance commitment.

1.2 Methodology

In an important contribution, Roberds (1987)
considers that promises may not always be kept and proposes the
probabilistic model also analyzed here. The author's model and
assumptions are very specific, and his method is not generalizable
to other applications. In another important contribution, Schaumburg and Tambalotti (2007) extended Roberds work to linear
quadratic models and apply their methods to a Barro-Gordon type of
monetary model without endogenous state variables. While their
methods are useful for models that are exactly linear quadratic,
most non-linear models can not be properly solved with Schaumburg and Tambalotti (2007). This is still the case if a
linear quadratic approximation is performed to the non-linear
model. As shown by Curdia et al. (2006), Debortoli and Nunes (2006) and Benigno and Woodford (2006), a correct linear-quadratic
approximation can in general be derived if one imposes the timeless
perspective assumption. The timeless perspective assumes that the
problem is initialized at the full commitment steady state and that
default never occurs. The loose commitment framework clearly
requires a departure from the timeless perspective. In such cases,
using the linear-quadratic approach with loose commitment is
inappropriate because one considers different assumptions and the
specification of the original model is violated. As discussed in
the above references, this problem would occur whenever the planner
can not achieve the first best allocation, for instance due to the
presence of distortionary taxes. Unlike Schaumburg and Tambalotti (2007), our methodology can be applied
to models that are not linear quadratic.

The tools for the analysis of time-inconsistent and
time-consistent policy are recent. The key reference for solving
time-inconsistent models is Marcet and Marimon (1998).
Klein and Ríos-Rull (2003) show how to solve for the
time-consistent policy with linear quadratic techniques. Klein et al. (2007) recognize that the techniques proposed in
Klein and Ríos-Rull (2003) do not deliver controlled accuracy
and propose a technique based on generalized Euler equations and a
steady state local analysis. Judd (2004) proposes
global approximation methods instead of steady state local
analysis. In the solution procedure, we use a global method and
generalized Euler equations taking the recent contributions of
Judd (2004) and Klein et al. (2007). Besides
the points presented in Judd (2004), employing a
global method is specially important here. For the solution to be
accurate, one needs to perform a good approximation both in
commitment and default states. These two states, and the
corresponding policy functions are not necessarily similar for one
to be a priori certain that a local approximation would
suffice.

Finally, we prove the recursiveness of the solution using the
tools of Marcet and Marimon (1998). We show how to solve
for linear and non-linear models, without and with endogenous state
variables, relying only on one fixed point. As a by-product, our
methodology can be used as a homotopy method to obtain the
time-consistent solution.

1.3 Literature Review

Unlike the reputational equilibria literature, as in Backus and Driffill (1985), we are not aiming at building
setups where a planner of a certain type resembles another type. We
aim at characterizing the solution of planners that can make
credible promises, but the commitment technology may become
inoperative when it is time to fulfill them.

Another related topic is the trigger strategies, as in Barro and Gordon (1983b). Our paper is not aimed at building
equilibria where private agents try to enforce a given equilibrium.
Such strategies are quite intricate and raise enforcement and
coordination issues.1 Hence, one can think that the planner
may not always be forced to fulfill its promises, as in the
loose commitment setting.

Flood and Isard (1989) consider a central bank
commitment to a rule with escape clauses. The rule does not
incorporate some important shocks affecting the economy. When such
shocks hit the economy, it may be better to abandon the rule. One
can interpret that our probability of default is their probability
of anomalous shocks. Another interpretation is that we consider
policymakers who are more rational, and do not leave important
shocks outside the commitment rule. In such interpretation, the
rule is always better and the planner only defaults if the
commitment technology becomes inoperative.

Persson et al. (2006), elaborating on an
earlier proposal of Lucas and Stokey (1983), suggest a
mechanism that makes the commitment solution to be time-consistent.
Each government should leave its successor with a carefully chosen
maturity of nominal and indexed debt for each contingent state of
nature and at all maturities. Even though such strategies do
eliminate the time-consistency problem, this structure of debt is
not observed in reality. The view of this paper is that at certain
points in time the commitment solution may be enforced, but in some
contingencies discretion is unavoidable. This paper will consider a
model with endogenous public good. Rogers (1989)
showed that in such case debt restructuring can not enforce the
commitment solution.

More importantly, most of the macroeconomics literature has
either followed a commitment or discretion approach. This paper
presents a general method that can potentially fill this gap. This
paper can characterize policy under the more realistic description
that some promises are fulfilled while others are not. These
methods can be directly applied to the dynamic political economy
literature, where different governments alternate in office, as in
Alesina and Tabellini (1990). Due to technical reasons
that this paper overcomes, such literature had always assumed a
discretion approach or avoided time-inconsistency issues.

2 The Model

The methods and frameworks described in this paper can be
applied to a wide set of dynamic optimization problems. Instead of
discussing the methods in an abstract way, we will show an
application to a fiscal policy problem. The model we are going to
use has been described for instance in Martin (2007).

A representative household derives utility from private
consumption , public consumption
and leisure . The household has 1 unit of time
each period that he can allocate between leisure and labor
. The household rents capital and supplies labor to a firm. Labor
and capital earnings are taxed at a rate
and
respectively.

Following Greenwood et al. (1988), the household can
also decide on the capital utilization rate .
Therefore, the amount of capital rented to firms will be
. We are also going to assume that the
depreciation rate of capital is a function of its utilization rate,
. In this model, we are going
to abstract from debt. Dealing with debt and commitment issues is a
topic that requires extensive treatment on itself and is beyond the
scope of this paper.2

where denotes the discount factor. There is
uncertainty in this economy because it is not know in advance
whether the planner will default or not. The households' first
order conditions (FOC) are:

(2)

(3)

(4)

The time-inconsistency of this problem appears in Eq. (2).
Today's household decisions depend on the expectation of future
variables. In particular, the contemporaneous capital accumulation
decision depends on future returns on capital. It is important to
note that Eq. (4) is stating that
the current capital tax is distortionary. If the planner would
raise capital taxes, households could choose a lower capital
utilization rate.3 This feature is important for our
model because if the planner does not keep his past promises and
the capital utilization is fixed, then the capital tax could be set
at an extremely high and implausible value. Martin (2007) showed that with fixed capital utilization,
and for plausible calibrations, an equilibrium under discretion
does not exist. There may be other reasons that inhibit the planner
to choose an extremely high capital rate when it defaults, we chose
this specification that guarantees the model to have a well defined
solution for the commitment and discretion case.

Total output is produced according to the
function
. Firms
operate in perfectly competitive markets. Hence, wages and interest
rates are given by:

(5)

(6)

The planner provides the public good , sets taxes
and ,
satisfying the balanced budget constraint:

(7)

Combining the households and governments' budget constraint one
obtains the feasibility constraint:

(8)

To make our problem simpler we can proceed with a number of
simplifications. Eq. (4) can be used to
express the capital utilization as a function of other
variables:

(9)

Similarly, using the household and government budget constraint,
private and public consumption can be expressed as functions:

(10)

(11)

Hence, the FOCs in Eqs. (2, 3) can
be written in a more compact form:

(12)

The vector of functions
depends on several variables,
where
is the
vector of contemporaneous control variables, is the state variable and
is the history of events up to
.4 Note that
and have
already been substituted in Eq. (12).

3 The Probabilistic Model

We will consider a model where a planner is not sure whether his
promises will be kept or not. As explained above, this uncertainty
can be due to several factors. For simplicity, we assume that these
events are exogenous and that in any period the economy will
experience default or commitment with given exogenous
probabilities. In Section 4,
we relax this assumption. Since it is indifferent whether it is the
same or a new planner who defaults and reoptimizes, we use the
terms "reelection", "new planner" and "default"
interchangeably.

To make matters simple, we abstract from any shock other than
the random variable describing default
() or commitment () in period
. It is a straightforward generalization to
include other sources of uncertainty, but the notation would be
harder to follow. More formally, suppose the occurrence of Default
or No Default is driven by a Markov stochastic process
with possible
realizations
, and let
be the set of possible histories
up to time t:

(13)

We only consider the histories
,
i.e. histories that start with a default on past promises. This is
because in the initial period there are no promises to be fulfilled
or equivalently the current government has just been settled.
Before turning to the planner's problem, we describe the problem of
individual agents.

3.1 Individual Agents and
Constraints

In Eq. (12) we wrote the
households' FOCs. These equations depend on future variables and
hence households need to form rational expectations using available
information. Given our institutional setting, households believe
the promises of the current planner, but consider that if a
different planner comes into play, then different policies will be
implemented and past promises will not be kept. As it is common in
the time-consistency literature, economic agents will take future
controls that can not be committed upon as functions of the state
variable, i.e.
where we use
the short notation
to denote
. The function
denotes the vector of policy
functions that rational agents anticipate to be implemented in
future periods.5Therefore, the constraint in Eq.
(12)
becomes:

(14)

where we use the short notation
to
denote
. Note that is a state variable and hence it is
understood that
,
.

3.2 The Planner

When default occurs, a new planner is appointed and it will be
taking decisions from that point onwards. Therefore, it is
convenient to separate all histories
with respect to the first time
when default occurs. This is because we want to know which
histories correspond to which planner. We now define the subset of
of histories where only commitment
has occurred up to time as:

(15)

and the subsets of histories where the first default occurs in
period ,

, if

(16)

,
if

By construction note that
is
a partition of the set
. Moreover, it can be seen that the
sets
and
are singletons.6
Therefore, in order to avoid confusion between histories and sets
of histories, we will refer to these singleton sets as
and
respectively.

In figure 1 we
show a more intuitive representation of the particular partition of
histories specified above, where we use the name of the unique
history ending in a given node to denote the node itself. White
nodes indicate when a new planner is settled (default has
occurred), while black nodes indicate the cases where the first
planner is still in power (no default has occurred). We can see
that in any period there is only one history
such that commitment has
always occurred in the past. Moreover, there is also only one
history
,
meaning that the first default occurred in period .
In our institutional setting, a new planner is then settled from
the node
onward and it will make its
choices over all the possible histories passing through the node
, that is the sets
.

Figure 1: Diagram of the possible histories

We will now write the problem of the current planner where to
simplify notation, and without loss of generality, we abstract from
the presence of constraints in the maximization problem:

(17)

where we are using the short notation
.
Equation (17) makes it
explicit that inside the maximization problem of the current
government there are other planners maximizing welfare during their
tenures. Given that
is
a partition of the set
, all the histories are
contemplated in our formulation. Since ,
, we can rewrite the probabilities for
in the
following way:

,

(18)

Substituting for these expressions into Eq. (17) and
collecting the common term in the summation, we obtain:

(19)

Since we are assuming that any future planner is also maximizing
we can define the value functions:

(20)

where it was made explicit that each planner assigns probability
one to its initial node. The value functions
summarize the happenings
after the node
. Since
for , the choices of future planners are independent
between themselves. This formulation is very general since one can
assume several institutional settings that the future planners will
face. For example, one can assume that some future planners have
full commitment while others do not. For simplicity we will assume
that all future planners face the same institutional settings which
at this stage we do not specify, thus we assume that
.7 Since all the histories
are already being maximized by other planners, it is equivalent to
consider that the initial planner maximizes over the single history
instead of
. We can therefore
rewrite the problem at period as:

(21)

We will now assume that the random variable is i.i.d. to further simplify the problem. It is
straightforward to generalize our formulation to Markov processes.
Also to simplify notation denote
and
,
which implies that:

(22)

(23)

With this formulation at hand we are ready to show that our
problem can be written as a saddle point functional equation
(SPFE), and that the optimal policy functions of the planner are
time-invariant and depend on a finite set of states.

3.2.1 The Recursive Formulation

Collecting results from the previous section, the problem of the
current planner is:

(24)

Due to the fact that we do have future controls in the
constraints through the term
, the usual Bellman equation is not satisfied.8 Building on the
results of Marcet and Marimon (1998), we show that
problems of this type can be rewritten as a SPFE that generalizes
the usual Bellman equation. This result is summarized in
proposition 1.

Proposition 1Problem (24) can be written
as saddle point functional equation as:

(25)

where

(26)

(27)

(28)

Proof. See the appendix.

Proposition 1 makes it clear that the current planner maximizes
utility of the representative agent subject to the constraints
, where the
latter is incorporated in . If there is no
commitment, the continuation of the problem is
. If the current promises
will be fulfilled, then the continuation of the problem is
, and promises
are summarized in the co-state variable
. The co-state variable is
not a physical variable and the policymaker always faces the
temptation to set it to zero. Also note that in our problem only
the first constraint contained in Eq. (12) contains future
control variables. Therefore, only the first element of the vector
needs to be included as a co-state
variable. The optimal policy functions of such problem are time
invariant and depend on a finite number of states, as proposition 2
describes.9

Proposition 2The solution of problem (24) is a
time invariant function with state variables
, that is to say:

(29)

Proof. See the appendix.

3.3 Equilibrium

In the institutional setting built in Eq. (24), we
only assume that all planners from period onward
will face the same problems. From now on, we also assume that all
future planners face the same institutional setting as we specify
in period 0. In other words, we specify their problems in the same
way as the problem of the planner in period 0. Thus we can use the
following definition of equilibrium.

Definition 1A
Markov Perfect Equilibrium where each planner faces the same
institutional setting must satisfy the following
conditions.

Given and , the
sequence solves problem (24);

The value function
is such that
;

The policy functions
solving problem (24) are
such that
.

The second part of the definition imposes directly that the
problem of the initial and future planners must be equal. When a
planner comes to office, he has not previously made any promise and
therefore the co-state variable is reset to zero. While a planner
is in office, he makes promises, and faces the temptation to
deviate and reoptimize. In other words, the multiplier encoding the
planner's promises is not a physical state variable and could
always be put to zero. We are assuming that such a deviation only
occurs with probability . The third part of the
definition imposes a consistency requirement in the constraints.
More precisely, we require the policy functions that agents expect to be implemented under default to
be consistent with the optimal policy function. We refer to the
notion of Markov Perfect Equilibrium because the function
only depends on the natural state
variables . Also, in this equilibrium neither the
planner nor individual agents desire to change behavior. Individual
agents are maximizing and their beliefs are correct. The planner,
taking as given and , is
also maximizing.

3.4 Solution Strategy

The previous propositions showed that the problem is recursive.
But at first sight, solving this problem looks daunting. The policy
functions and the value function appear in the constraints and the
objective function. One could try to guess
and , solve the problem, update
and , and
iterate until convergence. Such procedure would imply solving three
fixed points, one for the problem itself and two for and . In addition, this
problem faces simultaneously all the difficulties present in the
commitment and discretion literature. One has to include the
lagrange multiplier as a state variable, and the derivatives of
policy functions matter for the solution.

In this section, we discuss how to solve the problem in an
easier way. We use the FOCs of the associated lagrangian
formulation. Our generic problem is:

(30)

where and are defined by
Eqs. (27, 28) respectively.

Details on the FOCs can be found in the appendix. The term
appears in the FOCs, because the
current planner will try to influence future planners. The value
function
summarizes the
welfare that agents will achieve with a planner appointed at
. From the perspective of the planner
appointed at , the state variables can not be changed. Nevertheless, from the
perspective of the current planner, who is in charge at period
, is not given
and can be set strategically.10

The FOCs expressed in Eqs. (46-48) allows us to
solve for the optimal policy. As described in Definition 1, we are
particularly interested in the formulation where future planners
face the same problem as the current planner, i.e. where
and hence
. We will show a
solution method that only relies on solving one fixed point. To
obtain the derivative we can use envelope
results, which are summarized in result 1.

Result 1Using envelope results it follows that:

(31)

where all variables are evaluated using the optimal policy of a
planner appointed in period , given the state
.

Result 1 uses the fact
that all the planners are maximizing the same function, which
allows the use of envelope principles.11 It is important to
note that in Eq. (31) all the
variables are evaluated with the optimal policy of a newly elected
government.

By Definition 1, the policy
functions that the current and future planners implement are equal.
If we use the envelope result to substitute
, the FOCs only depend
on the functions
and . Using Definition 1 and Proposition 1 we know that
, which also
considerably simplifies the problem. We use a collocation method to
solve for the optimal policy functions. Hence, using Result 1,
Proposition 1 and 2, we can solve the problem relying on only one
fixed point. Note that, unlike Schaumburg and Tambalotti (2007), we have endogenous state
variables. Only in this case the derivative of the policy and value
function appear in the FOCs, creating further difficulties.

We want to stress that in our framework the global solution
methods proposed in Judd (1992) and Judd (2004) are more appropriate than local approximations.
Besides the arguments presented by Judd, there are other reasons
specific to our problem. The value function derivative, the levels
and derivative of policy under default are present in the FOCs. The
allocations under default and commitment are not likely to be
similar. Demanding for a local approximation to deliver a good
approximation in distant points to levels, derivatives and value
functions is quite demanding.

The linear quadratic approximation proposed in Benigno and Woodford (2006) is only valid in a timeless
perspective. The timeless perspective assumes that initial
commitments are equal to the steady-state commitment. There are
several reasons that make the timeless perspective approach
inappropriate in our framework. Firstly, we consider that
commitments may be broken and consequently we need to focus on
transition dynamics at that point. Secondly, our model does not
have a deterministic steady state point around which one can take
an approximation. Indeed, shutting down uncertainty completely
changes the problem. Thirdly, under discretion the allocations can
be very far from the commitment steady-state. Our method is more
suitable and it is also simpler. Even for an exactly linear
quadratic model, Schaumburg and Tambalotti (2007) only
solved a model with no endogenous state variables, and suggested a
procedure to handle endogenous state variables relying on three
fixed points. The method presented here relies on only one fixed
point in policy functions.

Besides these considerations, there is a crucial drawback of
applying the linear-quadratic timeless perspective approach to
study problems with loose commitment settings, as suggested
by Schaumburg and Tambalotti (2007). This point was
already discussed in the introduction and its methodological
discussion.

3.5 Results

In order to proceed to the numerical solution, we specify a
per-period utility function:

(32)

and a depreciation function:

(33)

We use a standard calibration for an annual model of the US
economy. Table 1 summarizes the
values used for the parameters. The parameters and imply that in steady
state the capital utilization rate is about
0.8, and the depreciation rate
is about 0.08.

Table 1: Parameter values

Parameter

Value

Description

β

.96

Discount factor

Φc

.285

Weight of consumption vs. leisure

Φg

.119

Weight of public vs. private consumption

θ

.36

Capital share

χ0

.171

Depreciation function parameter

χ1

1.521

Depreciation function parameter

Table 2
presents the long run average for several variables, and across
different parameterizations of . The column with
and correspond to
full commitment and full discretion respectively. In the full
commitment model, the capital tax is zero, a result common in the
optimal taxation literature with full commitment. With full
discretion, the capital tax is roughly 19%. In the discretion
model, once capital has been accumulated the government has a
temptation to tax it. Due to the possibility of changing the
capital utilization rate, capital is not an entirely fixed factor
of production that can be heavily taxed. The average capital
utilization rate does not change much with . But
as we will discuss later, if private agents are surprised with
higher than expected capital taxes, then the capital utilization
rate is lowered.

Table 2: Average Values

Variable

1.00

0.75

0.50

0.25

0.00

k

1.122

0.947

0.899

0.880

0.870

λ

-0.536

-0.177

-0.080

-0.030

0.000

g

0.093

0.076

0.072

0.070

0.069

c

0.196

0.216

0.220

0.222

0.224

y

0.378

0.368

0.364

0.363

0.362

τk

0.000

0.131

0.163

0.178

0.187

τl

0.384

0.251

0.218

0.203

0.191

l

0.233

0.245

0.248

0.250

0.250

u

0.798

0.799

0.801

0.800

0.799

Note: The values refer to long run averages.

In this model, governments cannot issue debt and have to balance
their budgets every period. In Chamley (1986), there
is a big incentive to tax capital very highly in earlier periods to
obtain large amounts of assets and eliminate distortionary taxation
in later periods.12 By imposing a balanced budget,
higher capital taxation revenues have to be used immediately. Even
though in our model the incentives to tax capital under discretion
are mitigated, it is still the case that capital taxes under
discretion are higher. For an example, where the reverse can happen
the reader is referred to Benhabib and Rustichini (1997)
and Klein et al. (2007).

Since capital taxes are lower under commitment, the level of
capital is higher. As a direct consequence of no capital tax
revenues, labor taxes need to be higher. Higher labor taxes induce
households to work less. Private consumption is lower in the full
commitment economy, while public consumption is higher. Obviously,
the allocations in terms of leisure, private and public consumption
are more efficient in the full commitment economy.

We now turn into commenting the loose commitment
settings. The main purpose of this paper is to provide a
theoretical basis for this kind of models. Providing a definitive
answer on the probability or empirically
estimating these models is beyond the scope of this paper.
Nevertheless, we can provide some evidence on the probability
. A probability of reelection implies an expected tenure of . A
value of 0.75 corresponds to a planner being in office for 4 years
on average. A calibration based on the political history of the US
implies a value of 0.8, while the political history of Italy would
imply a calibration around 0. The numbers above are excluding the
possibility that there were no broken promises during the tenure,
therefore a value of 0.75 might be considered an upper bound.
Rallings (1987) tried to obtain a measure of how
many manifesto pledges were actually implemented. Some of these
pledges often reveal political options, such as the composition of
expenditures, and may not always be related to time-inconsistency
issues. The average number reported by the author is 0.63 and 0.71
for Britain and Canada respectively. Arguably, these estimates may
be considered higher than the actual .
Ideological promises are easier to fulfill than promises where a
temptation to default actually exists. While Rallings (1987) estimates includes both types of promises,
the measure only refers to the latter.
Obviously, the measure depends on the country
being studied and the specific policy and institutional settings
involved.

In our analysis, all variables seem to be much closer to the
discretion solution rather than to the commitment solution. If the
probability of committing is 0.75, average allocations only move
about 31% of the distance from discretion to commitment. Table
3 computes this
value for all the allocations. The other way to interpret the table
is that small reductions in the probability from
the full commitment solution have dramatic effects. It may be
expected that decreasing the probability of commitment from 1 to
0.75 would lead allocations to move 25% of the difference between
commitment and discretion. But in fact, the absolute drop in
capital is 69% of the difference between full commitment and
discretion.

We now describe the transition dynamics. Figure 2 plots the
average path during the first 25 quarters, initializing capital at
its steady state value under discretion and considering that no
promises had been made. Since the economy starts with a relatively
low level of capital, the utilization rate of capital is relatively
high. This occurs despite the fact that capital taxes are high in
early periods as commitment starts to build. In initial periods,
since capital taxes are relatively high, labor taxes are lower. As
a consequence, labor is higher in initial periods. As time evolves,
the picture confirms the results of table 2, since for later
periods the variables are relatively closer to the discretion
path.

We should finally comment on the path of the lagrange
multiplier. This variable may not have economic interest per se,
since it is unobservable. We should nevertheless mention that as
all other variables, the lagrange multiplier is also much closer to
the discretion steady state of 0. This suggests that a local
approximation around the full commitment, as performed in the
timeless perspective approach may be a poor approximation to
loose commitment frameworks. Also our analysis suggests that
characterizing allocations in a full commitment analysis may be
less realistic than in a full discretion approach.

So far the analysis has only referred to average paths. We now
analyze what happens in the specific periods when governments
renege on their past promises. For this purpose, in figure 3, we plot the
paths followed in a given history. We consider the history where by
chance a new planner is reappointed every four years. When default
occurs the planner breaks the promise of a low capital tax. Capital
is a relatively inelastic tax base and hence capital taxes are
increased. Capital accumulation is almost unchanged but households
immediately react by decreasing the capital utilization rate. Since
there are more capital tax revenues, labor taxes are cut, leading
to more labor. Overall, output increases mainly fueled by the
increase in labor input.

Our analysis can be related to the literature on political
cycles, as described for instance in Drazen (2000).
The empirical analysis of political cycles mentions that output or
private consumption do not move much with the political cycle. In
our model, both output and private consumption are not found to
move much in relative terms. It is argued that policy instruments
are more affected by the political cycle. Our model has the same
prediction. Public consumption, capital taxes and labor taxes are
the policy instruments and these face greater variations in
relative terms. This empirical literature has also found that at
the end of a tenure output and private consumption are not higher
than average, a feature also predicted in our model. It is also
found that at the end of the tenure there are not significant tax
cuts leading to less tax revenues, which also conforms with our
model. There is some evidence that public expenditure is increased
before the elections, being the evidence stronger for government
transfers. This feature is also replicated in our model, public
consumption is higher as the tenure evolves. It is not our aim to
match political cycles, because electoral competition is absent
from the model. But our simple model does not contradict the
empirical evidence found in that literature.

3.6 Welfare Calculations

In this section, we turn to measure the welfare implications of
building commitment. In our framework, this means considering the
welfare gains of increasing . Consider two
regimes, an alternative regime (a) and benchmark regime (b). The
life-time utility of the representative agent
in both regimes is given by:

(34)

where
is the
optimal allocation sequence in regime . The
expectation operator covers all the commitment and default states.
Define as the increase in private
consumption in the benchmark regime that makes households
indifferent between the benchmark and an alternative regime. More
formally is implicitly defined as:

(35)

For the calculations that follow we considered the benchmark
regime to be the full discretion case and we initialized capital at
the steady state prevailing when .13 The
welfare improvement from discretion to commitment is equivalent to
an increase in private consumption of 3.65%. Figure 4 shows the
welfare gains for different probabilities of commitment. When the
probability of default increases from 0 to 0.25, only 13% of the
benefits of commitment are achieved. We plot the relative welfare
gains as a function of in figure 4. The
function is convex suggesting that increasing from low to intermediate levels results in relative small
welfare gains. Most of the gains from enhancing commitment can only
be achieved when is already high. In figure
5, we
plot the relative welfare gains as a function of the expected time
before a default occurs (). In this metric
the welfare gains function is concave. The welfare gains per unit
of time of moving from 1 to 2 years are higher than the gains of
moving from 2 to 3 years. This result may seem more intuitive,
since as the expected commitment period increases there are less
welfare gains to be achieved.

In a related work on optimal monetary policy, Schaumburg and Tambalotti (2007) found qualitatively different
results. First, allocations move linearly in the probability
. For instance, when moves from 1 to 0.75 inflation goes 25% of the distance
towards discretion. Secondly, most of the welfare gains are
achieved at low levels of commitment. In other words, the welfare
is always concave regardless of the metric used. When is 0.75, about 90 of the welfare gains from commitment
are obtained. Comparing absolute welfare measures in our model and
theirs is unclear, but the welfare gains of moving from discretion
to commitment are also much higher in their model.14

One possible reason for the different results obtained here and
in Schaumburg and Tambalotti (2007) is that fiscal and
monetary policy are simply different in this respect. We have
investigated other potential sources of differences between our
results and theirs. First, figure 5 suggests that
some differences could occur if one would change the time period of
the model. The original calibration used in Schaumburg and Tambalotti (2007) is quarterly. We also tried an
annual calibration of their model, and the results do not change
qualitatively. Another potential difference relates to the role of
endogenous state variables. An endogenous state variable, provides
the planner with an additional instrument to influence future
decisions. Therefore, it may be expected that in a model with
endogenous state variables, the benefit of adding commitment is
smaller. While our model has one endogenous state variable their
model has none. To test this hypothesis we also considered their
model with an hybrid Phillips curve as in Galí and Gertler (1999). The results do move slightly in the
direction that we predict but a significant difference between
their model and ours remains. Our results suggest that there may be
an important difference between monetary and fiscal policy in this
respect.

3.7 Discussion on Monetary and Fiscal
Policy Commitment

In the early 70's economics and policymakers were still not
aware of the importance of commitment and its use in policy design.
The commitment in both fiscal and monetary policy were low at that
time. After the seminal contributions in the late 70's and early
80's regarding time-inconsistency, there has been a concern to
increase institutional commitment. The reforms to increase monetary
policy commitment were far more intense than the ones relative to
fiscal policy. In fact, we observed a pattern of inflation that is
consistent with low commitment in the 70's and high commitment
today. In the 70's, the inflation level was much closer to the
steady state level of discretion than the one of commitment.
Nowadays, the opposite is true.

The patterns in fiscal policy are somehow different. We have not
observed institutional reforms aimed at increasing commitment
comparable to the ones in monetary policy. Also, as discussed in
Klein and Ríos-Rull (2003), the level of taxes both in the
70's and today are much closer to the full discretion prediction
than to the commitment one. This evidence suggests that fiscal
policy commitment was low in the 70's and is still low today, while
monetary policy commitment was low in the 70's and is high
today.

Combining our results with those of Schaumburg and Tambalotti (2007), may shed light on this issue.
It seems that when commitment is low, the welfare gains of more
commitment are higher in monetary policy than in fiscal policy.
Also, it may be very hard to implement the full commitment
solution, since defaults are always possible and flexibility is
necessary. While most welfare improvements can be achieved at
intermediate levels of commitment in monetary policy, the same is
not true for fiscal policy.

It may also be argued that it may be easier to achieve high
levels of commitment in monetary policy than in fiscal policy. The
turnover in presidents is higher than in central bank governors.
More importantly, it may be difficult to establish a fiscal
authority with full commitment, because such an institution would
have to rule out democratic choices as they violate the commitment
plan. The intuition that achieving high commitment levels is easier
in monetary policy rather than in fiscal policy together with our
results may also help in explaining the data.

4 Endogenous Probability Model

We are finally going to consider an extension where the
probability of defaulting depends on the states of the economy.
Since capital is the only natural state variable in the economy and
all allocations depend on capital, we will consider that the
probability of defaulting today depends on the current capital
stock. The planner and households will consider that the
probability of commitment in the next period is
instead of . Following the steps in earlier sections, the objective
function of the planner becomes:

(36)

As before, the probability of being in charge in the first
period is 1. The special term in the
objective function does not induce any time-inconsistency problem
because is predetermined.15

Households understand that the probability of committing depends
on aggregate capital. A single household can only decide his own
single capital accumulation. This means that each household is
atomistic and takes the aggregate capital stock as given.
Therefore, as it seems reasonable, the individual household capital
accumulation decision does not incorporate the effect in the
commitment technology.16 Hence, the constraints that the
planner faces are:

(37)

We need to prove that this setting can also be written as a
SPFE. This result is done in Proposition 3, and
details are available in the appendix.

Proposition 3The problem of a planner maximizing Eq. (36)
subject to Eq. (37) can be written
as saddle point functional equation.

Proof. See the appendix.

With proposition 3 at
hand, it then follows that the solution to the problem is a
time-invariant function. In the appendix, we also describe the FOCs
and some simplifications that are very useful for computational
work. In comparison with the exogenous probability case, the FOCs
with respect to all variables except to capital remain unchanged.
In the FOC with respect to capital, some new terms appear
reflecting that the commitment probability can be influenced.
Unlike households, the planner does not take aggregate capital as
given. One extra term refers to the appearance of
in the constraints of households.
The other term refers to the expected change in utility, induced by
the change in the commitment probability. This is captured by the
term
. If
capital is increased, the commitment probability is increased by
. This increases the chances of
the current planner to obtain tomorrow's continuation value under
commitment
. Nevertheless, it
decreases the chances of obtaining tomorrow's continuation value
under discretion
.

This model raises an extra difficulty in terms of computational
work. As explained above, the level of the value function appears
in the FOCs. Hence, one needs to approximate the value function as
well. The exogenous probability model could be solved as one fixed
point. The endogenous probability model needs to be solved as two
fixed points.

In what follows, we will consider a probability function such
that when capital is higher there is a higher probability of
commitment. This assumption could be justified on political economy
grounds. More capital implies more output and a higher probability
of reelection. We will consider the following probability
function:

(38)

The parameter is a normalization such
that
. The higher is , the easier it is for the planner to influence its
reelection probability. In the case of , the
probability is always constant. We can use a homotopy from the
model in section 3 to this model by
changing from 0 to the desired value. We chose
and to
be equal to the average capital allocation when . Our normalization of
allows us to directly compare the results with the probabilistic
model when .

Results are presented in table 4. In the
endogenous probability model, capital is now higher. Since the
probability of commitment is increasing in capital, the planner has
a further motive to accumulate capital. The incentives to
accumulate more capital need to be provided by the planner.
Households by themselves do not strategically increase their
capital in order to increase the commitment probability. In order
to make households accumulate more capital, the planner mainly
reduces capital taxes. Since more commitment is achieved, average
allocations move towards the full-commitment equilibrium.

In the endogenous probability model the welfare gain relative to
discretion is 2.6%. This value is higher than the welfare gains
obtained in the benchmark case of . One
reason is that the commitment probability is higher. The other
reason is that the welfare gains function is convex in . A varying probability around a mean may therefore induce
some additional welfare gains. In a political economy
interpretation, our model would suggest that governments accumulate
more capital to be reelected; and this is a good policy since it
reduces political turnover thus increasing the commitment
probability.

Table 4: Endogenous Probability - Average Values

Variable

π = 0.5

End. Prob.

k

0.899

0.932

λ

-0.080

-0.064

g

0.072

0.082

c

0.220

0.209

y

0.364

0.365

τk

0.163

0.153

τl

0.218

0.268

l

0.248

0.246

u

0.801

0.790

0.500

0.738

5 Conclusions

The time-consistent and time-inconsistent solutions are to some
extent unrealistic. This paper tried to characterize optimal policy
in a setting where some promises are fulfilled while others are
not. One interpretation of such setting is based on political
turnover, where governments make promises but may be out of power
when it is time to fulfill them. Alternatively, governments may
make promises but when certain events arise the commitment
technology breaks. This framework can also be thought as providing
an optimal policy prescription knowing that at a later date policy
is going to be revised.

From the methodological point of view, our contribution is to
show a solution technique for problems of loose commitment
with the following main features. First, it can be applied to a
wide class of non-linear models, with or without endogenous
state-variables keeping the model's micro-foundations structure
intact. While there were other works on similar loose
commitment settings, such methods could not be used in the
standard non-linear macro models. For instance, the fiscal policy
problem described here has to be solved with the methods developed
in this paper. Second, building on the results of Marcet and Marimon (1998), we proved that the solution to our
problem is recursive. Third, we implemented an algorithm which is
relatively inexpensive, and makes use of global approximation
techniques which are pointed out in the literature as more
reliable. Finally, as a by-product, our procedure can be used as a
homotopy method to find the time-consistent solution.

We show that in the optimal taxation model under loose
commitment, average allocations seem to be closer to the
time-consistent solution. We have also characterized the economic
consequences of reneging on promises. Our results suggest that when
promises are not fulfilled capital taxes will be raised and labor
taxes will be lowered. As a consequence, the capital utilization
rate drops and labor input increases. We then showed that several
features of the model are in accordance with some empirical results
on political cycles.

We then considered that the probability of committing would be a
function of the endogenous state variables. In such a model, the
government would try increase the commitment probability through
the state variables. The intuition for such results is that having
more commitment is welfare improving, therefore the government
tries to increase commitment.

Regarding welfare, we find that for an upper bound of the
probability of commitment around 0.75, most of the gains from
commitment are not achieved. While the welfare gains are a concave
function of the expected time before a default, they are a convex
function of the probability of commitment. These results are
different from those obtained in the literature on monetary policy.
This may explain the observation that more effort has been devoted
to building commitment in monetary rather than fiscal policy.

The methods provided in this paper are general and can be
applied to a variety of macroeconomic problems. The setups
formulated here can be easily brought into the dynamic political
economy literature. This literature has commonly abstracted from
the presence of time-inconsistency or assumed a discretion
approach. Our paper allows consideration of the more realistic
setting in which governments can fulfill promises when they are
reelected. This paper also sets up the base for addressing problems
where different governments do not face the same institutional
settings or disagree on policy objectives. Finally, the
applications of our methodology are not restricted to optimal
policy problems. Indeed, it can be used in many dynamic problems
where commitment plays an important role, like the relationship
between firms and their customers and shareholders, or in other
principal-agent problems.

A Proofs

Proof. of Proposition 1

Drop history dependence and define:

Our problem is thus:

(39)

which fits the definition of Program 1 in Marcet and Marimon (1998). To see this more clearly note that our
discount factor is and we have no
uncertainty. Since
is a singleton, we have
previously transformed our stochastic problem into a non-stochastic
problem. Therefore, we can write the problem as a saddle point
functional equation in the sense that there exists a unique
function satisfying

(40)

where

(41)

or in a more intuitive formulation define:

(42)

and the saddle point functional equation is:

(43)

Proof. of Proposition 2: Using
Proposition 1, this proof follows trivially from the results of
Marcet and Marimon (1998).

Proof. of Proposition 3

Define an additional variable , The law of
motion for is:

(44)

with
. The problem of the planner can
then be rewritten as:

(45)

Using similar redefinitions as in the proof of proposition
1 the
result follows. The condition
signals that a new planner is in
charge. Eq. (44)
is still valid because it only refers to the evolution of
through commitment states. Finally,
note that in terms of notation
.17

B First Order Conditions

B.1 Probabilistic Model

B.2 Endogenous Probability Model

To simplify the problem, it is useful to multiply the second
constraint by , which does not change the
solution. Set up the Lagrangian:

(49)

The FOCs are:

(50)

(51)

(52)

(53)

(54)

To obtain Eqs. (50,
51) one has to
divide the original FOC by
, and use Eq. (53).19 One
can solve Eq. (54) forward and
obtain:

(55)

This equation states that
is equal to times the value function starting at period t+1. Note
that it represents the value function when past promises are kept,
because all the terms considered refer to commitment states.
Simplifying notation:

(56)

One can use the equation above to eliminate
in Eq. (51).

For the numerical work all the simplifications above are
convenient to reduce the number of equations in the system. Since
is eliminated from the problem,
this variable is not a state variable necessary for the numerical
approximation. Intuitively, at each point in time the planner that
is in charge only needs to know the current promises summarized by
, and the capital stock
. The probability of committing between
and is a bygone. The
probability of committing between and
,
, is not predetermined.

Figure 2: Average Allocations

Note: The figure plots for several values of π the average path across realizations. Capital is initialized at the discretion steady state. The lagrange multiplier is initialized at zero, considering that there were no previous promises in the first period.

Data for Figure 2 - Capital Utilization

Period

Default

Commitment

0.00

0.25

0.50

0.75

1.00

1

0.79940

0.79791

0.79690

0.80412

0.81616

0.83980

0.88712

2

0.79940

0.79791

0.79725

0.80301

0.81207

0.82970

0.86577

3

0.79940

0.79791

0.79755

0.80241

0.80937

0.82195

0.84789

4

0.79940

0.79791

0.79780

0.80203

0.80750

0.81579

0.83294

5

0.79940

0.79791

0.79802

0.80170

0.80614

0.81132

0.82049

6

0.79940

0.79791

0.79821

0.80156

0.80510

0.80825

0.81014

7

0.79940

0.79791

0.79838

0.80135

0.80425

0.80570

0.80157

8

0.79940

0.79791

0.79852

0.80107

0.80361

0.80369

0.79453

9

0.79940

0.79791

0.79864

0.80083

0.80309

0.80236

0.78877

10

0.79940

0.79791

0.79875

0.80069

0.80268

0.80105

0.78411

11

0.79940

0.79791

0.79884

0.80055

0.80230

0.80013

0.78038

12

0.79940

0.79791

0.79891

0.80044

0.80203

0.79963

0.77744

13

0.79940

0.79791

0.79898

0.80033

0.80175

0.79901

0.77518

14

0.79940

0.79791

0.79904

0.80022

0.80142

0.79858

0.77350

15

0.79940

0.79791

0.79909

0.80008

0.80106

0.79826

0.77231

16

0.79940

0.79791

0.79913

0.79997

0.80076

0.79791

0.77153

17

0.79940

0.79791

0.79917

0.79987

0.80049

0.79803

0.77110

18

0.79940

0.79791

0.79920

0.79985

0.80029

0.79792

0.77096

19

0.79940

0.79791

0.79923

0.79980

0.80017

0.79794

0.77107

20

0.79940

0.79791

0.79925

0.79978

0.80009

0.79784

0.77139

21

0.79940

0.79791

0.79927

0.79979

0.80006

0.79795

0.77187

22

0.79940

0.79791

0.79929

0.79979

0.80002

0.79774

0.77248

23

0.79940

0.79791

0.79931

0.79970

0.79989

0.79843

0.77321

24

0.79940

0.79791

0.79932

0.79968

0.79993

0.79869

0.77402

25

0.79940

0.79791

0.79933

0.79973

0.79997

0.79846

0.77489

Data for Figure 2 - Labour

Period

Default

Commitment

0.00

0.25

0.50

0.75

1.00

1

0.25035

0.23282

0.24999

0.25249

0.25700

0.26573

0.28218

2

0.25035

0.23282

0.25004

0.25061

0.25314

0.26017

0.27683

3

0.25035

0.23282

0.25008

0.25011

0.25124

0.25624

0.27196

4

0.25035

0.23282

0.25012

0.24993

0.25027

0.25363

0.26759

5

0.25035

0.23282

0.25015

0.24992

0.24973

0.25153

0.26369

6

0.25035

0.23282

0.25018

0.24964

0.24942

0.24978

0.26023

7

0.25035

0.23282

0.25020

0.24982

0.24923

0.24879

0.25716

8

0.25035

0.23282

0.25022

0.24990

0.24901

0.24808

0.25444

9

0.25035

0.23282

0.25024

0.24981

0.24890

0.24735

0.25204

10

0.25035

0.23282

0.25025

0.24971

0.24878

0.24713

0.24990

11

0.25035

0.23282

0.25027

0.24973

0.24875

0.24677

0.24801

12

0.25035

0.23282

0.25028

0.24969

0.24865

0.24634

0.24632

13

0.25035

0.23282

0.25029

0.24970

0.24866

0.24630

0.24482

14

0.25035

0.23282

0.25029

0.24972

0.24874

0.24616

0.24348

15

0.25035

0.23282

0.25030

0.24976

0.24876

0.24601

0.24230

16

0.25035

0.23282

0.25031

0.24971

0.24873

0.24603

0.24124

17

0.25035

0.23282

0.25031

0.24971

0.24870

0.24564

0.24029

18

0.25035

0.23282

0.25032

0.24960

0.24862

0.24567

0.23945

19

0.25035

0.23282

0.25032

0.24966

0.24853

0.24561

0.23869

20

0.25035

0.23282

0.25033

0.24961

0.24848

0.24562

0.23802

21

0.25035

0.23282

0.25033

0.24959

0.24845

0.24548

0.23743

22

0.25035

0.23282

0.25033

0.24961

0.24845

0.24570

0.23689

23

0.25035

0.23282

0.25033

0.24973

0.24858

0.24500

0.23642

24

0.25035

0.23282

0.25033

0.24961

0.24834

0.24511

0.23599

25

0.25035

0.23282

0.25034

0.24952

0.24834

0.24553

0.23562

Data for Figure 2 - Labour Tax

Period

Default

Commitment

0.00

0.25

0.50

0.75

1.00

1

0.19091

0.38354

0.19092

0.19645

0.20014

0.20703

0.22218

2

0.19091

0.38354

0.19092

0.20094

0.20915

0.22016

0.23610

3

0.19091

0.38354

0.19092

0.20198

0.21329

0.22895

0.24837

4

0.19091

0.38354

0.19092

0.20232

0.21515

0.23425

0.25925

5

0.19091

0.38354

0.19092

0.20221

0.21605

0.23866

0.26898

6

0.19091

0.38354

0.19092

0.20287

0.21647

0.24259

0.27774

7

0.19091

0.38354

0.19092

0.20233

0.21663

0.24440

0.28568

8

0.19091

0.38354

0.19092

0.20202

0.21694

0.24558

0.29292

9

0.19091

0.38354

0.19092

0.20216

0.21702

0.24714

0.29956

10

0.19091

0.38354

0.19092

0.20235

0.21715

0.24719

0.30567

11

0.19091

0.38354

0.19091

0.20225

0.21710

0.24780

0.31132

12

0.19091

0.38354

0.19091

0.20232

0.21724

0.24882

0.31654

13

0.19091

0.38354

0.19091

0.20225

0.21710

0.24864

0.32139

14

0.19091

0.38354

0.19091

0.20215

0.21676

0.24883

0.32589

15

0.19091

0.38354

0.19091

0.20202

0.21656

0.24908

0.33008

16

0.19091

0.38354

0.19091

0.20210

0.21654

0.24887

0.33398

17

0.19091

0.38354

0.19091

0.20205

0.21653

0.25008

0.33761

18

0.19091

0.38354

0.19091

0.20233

0.21666

0.24988

0.34100

19

0.19091

0.38354

0.19091

0.20216

0.21684

0.25009

0.34415

20

0.19091

0.38354

0.19091

0.20229

0.21694

0.24999

0.34709

21

0.19091

0.38354

0.19091

0.20235

0.21700

0.25046

0.34983

22

0.19091

0.38354

0.19091

0.20229

0.21699

0.24967

0.35237

23

0.19091

0.38354

0.19091

0.20195

0.21659

0.25205

0.35475

24

0.19091

0.38354

0.19091

0.20224

0.21723

0.25181

0.35696

25

0.19091

0.38354

0.19091

0.20249

0.21722

0.25038

0.35901

Data for Figure 2 - Capital Tax

Period

Default

Commitment

0.00

0.25

0.50

0.75

1.00

1

0.18702

-0.00007

0.18732

0.18399

0.17913

0.16942

0.14823

2

0.18702

-0.00007

0.18728

0.17959

0.17073

0.15795

0.13729

3

0.18702

-0.00007

0.18724

0.17857

0.16689

0.15035

0.12767

4

0.18702

-0.00007

0.18721

0.17827

0.16519

0.14593

0.11915

5

0.18702

-0.00007

0.18719

0.17842

0.16441

0.14213

0.11152

6

0.18702

-0.00007

0.18716

0.17772

0.16409

0.13858

0.10461

7

0.18702

-0.00007

0.18714

0.17833

0.16403

0.13706

0.09829

8

0.18702

-0.00007

0.18713

0.17870

0.16376

0.13610

0.09244

9

0.18702

-0.00007

0.18711

0.17858

0.16375

0.13465

0.08701

10

0.18702

-0.00007

0.18710

0.17839

0.16365

0.13477

0.08191

11

0.18702

-0.00007

0.18709

0.17851

0.16376

0.13425

0.07712

12

0.18702

-0.00007

0.18708

0.17846

0.16365

0.13325

0.07260

13

0.18702

-0.00007

0.18707

0.17854

0.16384

0.13352

0.06831

14

0.18702

-0.00007

0.18707

0.17866

0.16426

0.13338

0.06424

15

0.18702

-0.00007

0.18706

0.17882

0.16451

0.13312

0.06038

16

0.18702

-0.00007

0.18705

0.17874

0.16458

0.13340

0.05671

17

0.18702

-0.00007

0.18705

0.17881

0.16463

0.13214

0.05322

18

0.18702

-0.00007

0.18705

0.17852

0.16452

0.13235

0.04990

19

0.18702

-0.00007

0.18704

0.17870

0.16433

0.13217

0.04674

20

0.18702

-0.00007

0.18704

0.17856

0.16422

0.13224

0.04375

21

0.18702

-0.00007

0.18704

0.17850

0.16418

0.13175

0.04091

22

0.18702

-0.00007

0.18704

0.17856

0.16418

0.13257

0.03821

23

0.18702

-0.00007

0.18703

0.17893

0.16462

0.13001

0.03566

24

0.18702

-0.00007

0.18703

0.17862

0.16392

0.13020

0.03325

25

0.18702

-0.00007

0.18703

0.17836

0.16391

0.13167

0.03097

Data for Figure 2 - Output

Period

Default

Commitment

0.00

0.25

0.50

0.75

1.00

1

0.36167

0.37812

0.36141

0.36491

0.37104

0.38298

0.40592

2

0.36167

0.37812

0.36145

0.36340

0.36809

0.37919

0.40397

3

0.36167

0.37812

0.36148

0.36300

0.36658

0.37636

0.40179

4

0.36167

0.37812

0.36150

0.36285

0.36580

0.37438

0.39957

5

0.36167

0.37812

0.36153

0.36285

0.36536

0.37276

0.39742

6

0.36167

0.37812

0.36155

0.36261

0.36510

0.37139

0.39538

7

0.36167

0.37812

0.36157

0.36277

0.36494

0.37057

0.39350

8

0.36167

0.37812

0.36158

0.36284

0.36477

0.36998

0.39177

9

0.36167

0.37812

0.36159

0.36277

0.36467

0.36937

0.39020

10

0.36167

0.37812

0.36160

0.36269

0.36458

0.36916

0.38878

11

0.36167

0.37812

0.36161

0.36271

0.36454

0.36886

0.38750

12

0.36167

0.37812

0.36162

0.36267

0.36447

0.36850

0.38636

13

0.36167

0.37812

0.36163

0.36268

0.36448

0.36845

0.38533

14

0.36167

0.37812

0.36164

0.36270

0.36455

0.36833

0.38441

15

0.36167

0.37812

0.36164

0.36273

0.36457

0.36819

0.38360

16

0.36167

0.37812

0.36165

0.36269

0.36454

0.36819

0.38287

17

0.36167

0.37812

0.36165

0.36270

0.36452

0.36788

0.38223

18

0.36167

0.37812

0.36165

0.36260

0.36446

0.36789

0.38166

19

0.36167

0.37812

0.36166

0.36265

0.36439

0.36785

0.38115

20

0.36167

0.37812

0.36166

0.36261

0.36434

0.36784

0.38071

21

0.36167

0.37812

0.36166

0.36259

0.36432

0.36772

0.38031

22

0.36167

0.37812

0.36166

0.36261

0.36431

0.36789

0.37997

23

0.36167

0.37812

0.36166

0.36271

0.36442

0.36733

0.37967

24

0.36167

0.37812

0.36167

0.36261

0.36422

0.36740

0.37940

25

0.36167

0.37812

0.36167

0.36254

0.36422

0.36773

0.37917

Data for Figure 2 - Consumption

Period

Default

Commitment

0.00

0.25

0.50

0.75

1.00

1

0.22354

0.19581

0.22380

0.22149

0.21892

0.21401

0.20451

2

0.22354

0.19581

0.22376

0.22155

0.21912

0.21453

0.20564

3

0.22354

0.19581

0.22373

0.22159

0.21928

0.21490

0.20642

4

0.22354

0.19581

0.22371

0.22162

0.21941

0.21523

0.20692

5

0.22354

0.19581

0.22368

0.22166

0.21952

0.21542

0.20721

6

0.22354

0.19581

0.22366

0.22166

0.21962

0.21549

0.20733

7

0.22354

0.19581

0.22365

0.22170

0.21970

0.21562

0.20732

8

0.22354

0.19581

0.22363

0.22173

0.21975

0.21574

0.20721

9

0.22354

0.19581

0.22362

0.22175

0.21981

0.21577

0.20702

10

0.22354

0.19581

0.22361

0.22177

0.21985

0.21589

0.20676

11

0.22354

0.19581

0.22360

0.22178

0.21989

0.21594

0.20645

12

0.22354

0.19581

0.22359

0.22179

0.21991

0.21592

0.20610

13

0.22354

0.19581

0.22358

0.22181

0.21995

0.21599

0.20573

14

0.22354

0.19581

0.22358

0.22182

0.22000

0.21603

0.20533

15

0.22354

0.19581

0.22357

0.22184

0.22004

0.21604

0.20492

16

0.22354

0.19581

0.22357

0.22185

0.22007

0.21609

0.20451

17

0.22354

0.19581

0.22356

0.22186

0.22010

0.21601

0.20409

18

0.22354

0.19581

0.22356

0.22186

0.22012

0.21603

0.20367

19

0.22354

0.19581

0.22356

0.22186

0.22012

0.21602

0.20326

20

0.22354

0.19581

0.22355

0.22186

0.22013

0.21603

0.20285

21

0.22354

0.19581

0.22355

0.22186

0.22013

0.21599

0.20245

22

0.22354

0.19581

0.22355

0.22186

0.22013

0.21606

0.20207

23

0.22354

0.19581

0.22355

0.22188

0.22016

0.21584

0.20169

24

0.22354

0.19581

0.22355

0.22187

0.22013

0.21582

0.20133

25

0.22354

0.19581

0.22355

0.22187

0.22013

0.21593

0.20099

Data for Figure 2 - Government Consumption

Period

Default

Commitment

0.00

0.25

0.50

0.75

1.00

1

0.06854

0.09281

0.06853

0.07005

0.07145

0.07410

0.07938

2

0.06854

0.09281

0.06853

0.07022

0.07189

0.07499

0.08101

3

0.06854

0.09281

0.06853

0.07025

0.07205

0.07551

0.08234

4

0.06854

0.09281

0.06854

0.07026

0.07210

0.07578

0.08344

5

0.06854

0.09281

0.06854

0.07025

0.07212

0.07598

0.08437

6

0.06854

0.09281

0.06854

0.07027

0.07212

0.07615

0.08517

7

0.06854

0.09281

0.06854

0.07025

0.07212

0.07621

0.08587

8

0.06854

0.09281

0.06854

0.07025

0.07212

0.07623

0.08648

9

0.06854

0.09281

0.06854

0.07025

0.07211

0.07627

0.08703

10

0.06854

0.09281

0.06854

0.07025

0.07211

0.07625

0.08752

11

0.06854

0.09281

0.06854

0.07025

0.07211

0.07626

0.08797

12

0.06854

0.09281

0.06854

0.07025

0.07211

0.07629

0.08837

13

0.06854

0.09281

0.06854

0.07025

0.07211

0.07627

0.08873

14

0.06854

0.09281

0.06854

0.07024

0.07210

0.07627

0.08907

15

0.06854

0.09281

0.06854

0.07024

0.07209

0.07626

0.08937

16

0.06854

0.09281

0.06854

0.07024

0.07209

0.07625

0.08965

17

0.06854

0.09281

0.06854

0.07024

0.07209

0.07630

0.08991

18

0.06854

0.09281

0.06854

0.07025

0.07209

0.07628

0.09015

19

0.06854

0.09281

0.06854

0.07024

0.07209

0.07629

0.09036

20

0.06854

0.09281

0.06854

0.07024

0.07209

0.07628

0.09056

21

0.06854

0.09281

0.06854

0.07025

0.07210

0.07630

0.09075

22

0.06854

0.09281

0.06854

0.07024

0.07209

0.07625

0.09092

23

0.06854

0.09281

0.06854

0.07024

0.07208

0.07635

0.09107

24

0.06854

0.09281

0.06854

0.07024

0.07210

0.07634

0.09122

25

0.06854

0.09281

0.06854

0.07025

0.07209

0.07626

0.09135

Data for Figure 2 - Lambda

Period

Default

Commitment

0.00

0.25

0.50

0.75

1.00

1

0.00000

-0.53644

0.00000

-0.02212

-0.04067

-0.05136

-0.04421

2

0.00000

-0.53644

0.00000

-0.02740

-0.06003

-0.08733

-0.08467

3

0.00000

-0.53644

0.00000

-0.02910

-0.06895

-0.10977

-0.12182

4

0.00000

-0.53644

0.00000

-0.02860

-0.07330

-0.12853

-0.15599

5

0.00000

-0.53644

0.00000

-0.03188

-0.07535

-0.14519

-0.18746

6

0.00000

-0.53644

0.00000

-0.02928

-0.07628

-0.15303

-0.21646

7

0.00000

-0.53644

0.00000

-0.02770

-0.07756

-0.15801

-0.24319

8

0.00000

-0.53644

0.00000

-0.02841

-0.07808

-0.16425

-0.26783

9

0.00000

-0.53644

0.00000

-0.02937

-0.07868

-0.16475

-0.29052

10

0.00000

-0.53644

0.00000

-0.02889

-0.07854

-0.16718

-0.31141

11

0.00000

-0.53644

0.00000

-0.02922

-0.07917

-0.17107

-0.33063

12

0.00000

-0.53644

0.00000

-0.02892

-0.07865

-0.17059

-0.34831

13

0.00000

-0.53644

0.00000

-0.02841

-0.07736

-0.17135

-0.36455

14

0.00000

-0.53644

0.00000

-0.02778

-0.07645

-0.17174

-0.37947

15

0.00000

-0.53644

0.00000

-0.02819

-0.07646

-0.17118

-0.39316

16

0.00000

-0.53644

0.00000

-0.02793

-0.07647

-0.17577

-0.40572

17

0.00000

-0.53644

0.00000

-0.02931

-0.07704

-0.17494

-0.41723

18

0.00000

-0.53644

0.00000

-0.02851

-0.07781

-0.17629

-0.42778

19

0.00000

-0.53644

0.00000

-0.02912

-0.07814

-0.17555

-0.43744

20

0.00000

-0.53644

0.00000

-0.02942

-0.07858

-0.17727

-0.44628

21

0.00000

-0.53644

0.00000

-0.02913

-0.07837

-0.17375

-0.45437

22

0.00000

-0.53644

0.00000

-0.02743

-0.07659

-0.18290

-0.46177

23

0.00000

-0.53644

0.00000

-0.02887

-0.07945

-0.18197

-0.46853

24

0.00000

-0.53644

0.00000

-0.03012

-0.07934

-0.17603

-0.47470

25

0.00000

-0.53644

0.00000

-0.02963

-0.08050

-0.17818

-0.48034

Data for Figure 2 - Capital

Period

Default

Commitment

0.00

0.25

0.50

0.75

1.00

1

0.87335

1.12224

0.87335

0.87335

0.87335

0.87335

0.87335

2

0.87335

1.12224

0.87291

0.87624

0.88193

0.89292

0.91354

3

0.87335

1.12224

0.87253

0.87731

0.88676

0.90702

0.94838

4

0.87335

1.12224

0.87220

0.87790

0.88974

0.91730

0.97846

5

0.87335

1.12224

0.87192

0.87831

0.89176

0.92501

1.00437

6

0.87335

1.12224

0.87167

0.87869

0.89325

0.93070

1.02664

7

0.87335

1.12224

0.87146

0.87881

0.89440

0.93475

1.04573

8

0.87335

1.12224

0.87128

0.87908

0.89535

0.93785

1.06205

9

0.87335

1.12224

0.87113

0.87942

0.89608

0.94027

1.07597

10

0.87335

1.12224

0.87099

0.87966

0.89668

0.94201

1.08780

11

0.87335

1.12224

0.87088

0.87981

0.89716

0.94350

1.09782

12

0.87335

1.12224

0.87078

0.87997

0.89759

0.94466

1.10626

13

0.87335

1.12224

0.87069

0.88009

0.89793

0.94543

1.11334

14

0.87335

1.12224

0.87062

0.88021

0.89825

0.94613

1.11923

15

0.87335

1.12224

0.87055

0.88035

0.89862

0.94670

1.12411

16

0.87335

1.12224

0.87050

0.88051

0.89900

0.94713

1.12809

17

0.87335

1.12224

0.87045

0.88062

0.89934

0.94755

1.13132

18

0.87335

1.12224

0.87041

0.88073

0.89962

0.94765

1.13390

19

0.87335

1.12224

0.87038

0.88074

0.89983

0.94778

1.13591

20

0.87335

1.12224

0.87034

0.88080

0.89996

0.94785

1.13744

21

0.87335

1.12224

0.87032

0.88081

0.90005

0.94793

1.13857

22

0.87335

1.12224

0.87030

0.88080

0.90010

0.94789

1.13934

23

0.87335

1.12224

0.87028

0.88081

0.90015

0.94803

1.13983

24

0.87335

1.12224

0.87026

0.88093

0.90031

0.94763

1.14007

25

0.87335

1.12224

0.87025

0.88094

0.90027

0.94733

1.14011

Figure 3: Particular History: Default Every 4 Periods

Note: The figure plots for several values of π a particular history realization. In this history the default occurs every four periods. Capital is initialized at the discretion steady state. The lagrange multiplier is initialized at zero, considering that there were no previous promises in the first period.

Data for Figure 3 - Capital Utilization

Period

Default

Commitment

0.00

0.25

0.50

0.75

1

0.79940

0.79791

0.79690

0.80412

0.81616

0.83980

2

0.79940

0.79791

0.79725

0.80646

0.81479

0.83157

3

0.79940

0.79791

0.79755

0.81292

0.81747

0.82718

4

0.79940

0.79791

0.79780

0.82300

0.82360

0.82598

5

0.79940

0.79791

0.79802

0.82034

0.81506

0.80461

6

0.79940

0.79791

0.79821

0.82069

0.81383

0.80129

7

0.79940

0.79791

0.79838

0.82552

0.81663

0.80098

8

0.79940

0.79791

0.79852

0.83422

0.82287

0.80321

9

0.79940

0.79791

0.79864

0.82961

0.81446

0.78623

10

0.79940

0.79791

0.79875

0.82878

0.81330

0.78530

11

0.79940

0.79791

0.79884

0.83265

0.81617

0.78701

12

0.79940

0.79791

0.79891

0.84055

0.82246

0.79098

13

0.79940

0.79791

0.79898

0.83481

0.81413

0.77630

14

0.79940

0.79791

0.79904

0.83331

0.81301

0.77662

15

0.79940

0.79791

0.79909

0.83663

0.81592

0.77939

16

0.79940

0.79791

0.79913

0.84408

0.82224

0.78427

17

0.79940

0.79791

0.79917

0.83771

0.81394

0.77084

18

0.79940

0.79791

0.79920

0.83584

0.81285

0.77182

19

0.79940

0.79791

0.79923

0.83885

0.81578

0.77517

20

0.79940

0.79791

0.79925

0.84604

0.82211

0.78055

21

0.79940

0.79791

0.79927

0.83932

0.81384

0.76781

22

0.79940

0.79791

0.79929

0.83723

0.81277

0.76916

23

0.79940

0.79791

0.79931

0.84008

0.81570

0.77282

24

0.79940

0.79791

0.79932

0.84712

0.82204

0.77848

25

0.79940

0.79791

0.79933

0.84021

0.81378

0.76612

Data for Figure 3 - Labour

Period

Default

Commitment

0.00

0.25

0.50

0.75

1

0.25035

0.23282

0.24999

0.25249

0.25700

0.26573

2

0.25035

0.23282

0.25004

0.24593

0.25028

0.25908

3

0.25035

0.23282

0.25008

0.24040

0.24466

0.25349

4

0.25035

0.23282

0.25012

0.23560

0.23984

0.24872

5

0.25035

0.23282

0.25015

0.25481

0.25684

0.26055

6

0.25035

0.23282

0.25018

0.24787

0.25015

0.25470

7

0.25035

0.23282

0.25020

0.24204

0.24455

0.24975

8

0.25035

0.23282

0.25022

0.23701

0.23974

0.24549

9

0.25035

0.23282

0.25024

0.25611

0.25675

0.25774

10

0.25035

0.23282

0.25025

0.24895

0.25007

0.25232

11

0.25035

0.23282

0.25027

0.24296

0.24449

0.24771

12

0.25035

0.23282

0.25028

0.23779

0.23969

0.24373

13

0.25035

0.23282

0.25029

0.25683

0.25670

0.25619

14

0.25035

0.23282

0.25029

0.24956

0.25003

0.25100

15

0.25035

0.23282

0.25030

0.24347

0.24445

0.24658

16

0.25035

0.23282

0.25031

0.23823

0.23966

0.24275

17

0.25035

0.23282

0.25031

0.25723

0.25667

0.25532

18

0.25035

0.23282

0.25032

0.24989

0.25001

0.25026

19

0.25035

0.23282

0.25032

0.24376

0.24443

0.24594

20

0.25035

0.23282

0.25033

0.23847

0.23964

0.24221

21

0.25035

0.23282

0.25033

0.25745

0.25666

0.25484

22

0.25035

0.23282

0.25033

0.25007

0.25000

0.24985

23

0.25035

0.23282

0.25033

0.24391

0.24442

0.24559

24

0.25035

0.23282

0.25033

0.23861

0.23963

0.24190

25

0.25035

0.23282

0.25034

0.25757

0.25665

0.25457

Data for Figure 3 - Labour Tax

Period

Default

Commitment

0.00

0.25

0.50

0.75

1

0.19091

0.38354

0.19092

0.19645

0.20014

0.20703

2

0.19091

0.38354

0.19092

0.21441

0.21815

0.22446

3

0.19091

0.38354

0.19092

0.23104

0.23456

0.23999

4

0.19091

0.38354

0.19092

0.24689

0.24998

0.25422

5

0.19091

0.38354

0.19092

0.19646

0.20014

0.20737

6

0.19091

0.38354

0.19092

0.21465

0.21814

0.22435

7

0.19091

0.38354

0.19092

0.23138

0.23454

0.23969

8

0.19091

0.38354

0.19092

0.24728

0.24996

0.25389

9

0.19091

0.38354

0.19092

0.19645

0.20014

0.20753

10

0.19091

0.38354

0.19092

0.21477

0.21813

0.22426

11

0.19091

0.38354

0.19091

0.23157

0.23453

0.23949

12

0.19091

0.38354

0.19091

0.24748

0.24995

0.25367

13

0.19091

0.38354

0.19091

0.19645

0.20014

0.20761

14

0.19091

0.38354

0.19091

0.21484

0.21813

0.22420

15

0.19091

0.38354

0.19091

0.23167

0.23453

0.23937

16

0.19091

0.38354

0.19091

0.24760

0.24994

0.25354

17

0.19091

0.38354

0.19091

0.19645

0.20014

0.20765

18

0.19091

0.38354

0.19091

0.21488

0.21813

0.22416

19

0.19091

0.38354

0.19091

0.23173

0.23452

0.23930

20

0.19091

0.38354

0.19091

0.24766

0.24994

0.25346

21

0.19091

0.38354

0.19091

0.19645

0.20014

0.20767

22

0.19091

0.38354

0.19091

0.21490

0.21813

0.22414

23

0.19091

0.38354

0.19091

0.23176

0.23452

0.23926

24

0.19091

0.38354

0.19091

0.24769

0.24994

0.25342

25

0.19091

0.38354

0.19091

0.19645

0.20014

0.20769

Data for Figure 3 - Capital Tax

Period

Default

Commitment

0.00

0.25

0.50

0.75

1

0.18702

-0.00007

0.18732

0.18399

0.17913

0.16942

2

0.18702

-0.00007

0.18728

0.16557

0.16151

0.15352

3

0.18702

-0.00007

0.18724

0.14744

0.14457

0.13884

4

0.18702

-0.00007

0.18721

0.12906

0.12777

0.12484

5

0.18702

-0.00007

0.18719

0.18208

0.17926

0.17372

6

0.18702

-0.00007

0.18716

0.16354

0.16164

0.15787

7

0.18702

-0.00007

0.18714

0.14541

0.14470

0.14298

8

0.18702

-0.00007

0.18713

0.12713

0.12789

0.12862

9

0.18702

-0.00007

0.18711

0.18101

0.17934

0.17608

10

0.18702

-0.00007

0.18710

0.16240

0.16172

0.16027

11

0.18702

-0.00007

0.18709

0.14429

0.14477

0.14529

12

0.18702

-0.00007

0.18708

0.12605

0.12796

0.13075

13

0.18702

-0.00007

0.18707

0.18041

0.17938

0.17738

14

0.18702

-0.00007

0.18707

0.16177

0.16176

0.16161

15

0.18702

-0.00007

0.18706

0.14367

0.14482

0.14657

16

0.18702

-0.00007

0.18705

0.12546

0.12800

0.13194

17

0.18702

-0.00007

0.18705

0.18008

0.17940

0.17810

18

0.18702

-0.00007

0.18705

0.16143

0.16179

0.16235

19

0.18702

-0.00007

0.18704

0.14332

0.14484

0.14729

20

0.18702

-0.00007

0.18704

0.12514

0.12802

0.13261

21

0.18702

-0.00007

0.18704

0.17990

0.17941

0.17851

22

0.18702

-0.00007

0.18704

0.16123

0.16180

0.16277

23

0.18702

-0.00007

0.18703

0.14313

0.14485

0.14770

24

0.18702

-0.00007

0.18703

0.12496

0.12803

0.13298

25

0.18702

-0.00007

0.18703

0.17980

0.17942

0.17874

Data for Figure 3 - Output

Period

Default

Commitment

0.00

0.25

0.50

0.75

1

0.36167

0.37812

0.36141

0.36491

0.37104

0.38298

2

0.36167

0.37812

0.36145

0.35961

0.36587

0.37849

3

0.36167

0.37812

0.36148

0.35497

0.36136

0.37451

4

0.36167

0.37812

0.36150

0.35081

0.35734

0.37096

5

0.36167

0.37812

0.36153

0.36659

0.37092

0.37914

6

0.36167

0.37812

0.36155

0.36109

0.36577

0.37507

7

0.36167

0.37812

0.36157

0.35626

0.36126

0.37147

8

0.36167

0.37812

0.36158

0.35195

0.35726

0.36826

9

0.36167

0.37812

0.36159

0.36752

0.37086

0.37699

10

0.36167

0.37812

0.36160

0.36190

0.36571

0.37316

11

0.36167

0.37812

0.36161

0.35698

0.36121

0.36977

12

0.36167

0.37812

0.36162

0.35258

0.35722

0.36676

13

0.36167

0.37812

0.36163

0.36803

0.37082

0.37579

14

0.36167

0.37812

0.36164

0.36235

0.36568

0.37209

15

0.36167

0.37812

0.36164

0.35737

0.36119

0.36882

16

0.36167

0.37812

0.36165

0.35292

0.35719

0.36592

17

0.36167

0.37812

0.36165

0.36831

0.37080

0.37511

18

0.36167

0.37812

0.36165

0.36260

0.36566

0.37149

19

0.36167

0.37812

0.36166

0.35759

0.36117

0.36829

20

0.36167

0.37812

0.36166

0.35312

0.35718

0.36545

21

0.36167

0.37812

0.36166

0.36847

0.37079

0.37473

22

0.36167

0.37812

0.36166

0.36273

0.36565

0.37115

23

0.36167

0.37812

0.36166

0.35771

0.36116

0.36799

24

0.36167

0.37812

0.36167

0.35322

0.35717

0.36518

25

0.36167

0.37812

0.36167

0.36855

0.37079

0.37452

Data for Figure 3 - Consumption

Period

Default

Commitment

0.00

0.25

0.50

0.75

1

0.22354

0.19581

0.22380

0.22149

0.21892

0.21401

2

0.22354

0.19581

0.22376

0.22099

0.21862

0.21418

3

0.22354

0.19581

0.22373

0.22003

0.21786

0.21390

4

0.22354

0.19581

0.22371

0.21868

0.21672

0.21325

5

0.22354

0.19581

0.22368

0.21981

0.21903

0.21756

6

0.22354

0.19581

0.22366

0.21953

0.21872

0.21722

7

0.22354

0.19581

0.22365

0.21876

0.21795

0.21651

8

0.22354

0.19581

0.22363

0.21758

0.21680

0.21550

9

0.22354

0.19581

0.22362

0.21887

0.21909

0.21948

10

0.22354

0.19581

0.22361

0.21872

0.21877

0.21888

11

0.22354

0.19581

0.22360

0.21805

0.21800

0.21794

12

0.22354

0.19581

0.22359

0.21696

0.21684

0.21674

13

0.22354

0.19581

0.22358

0.21835

0.21913

0.22055

14

0.22354

0.19581

0.22358

0.21827

0.21880

0.21980

15

0.22354

0.19581

0.22357

0.21766

0.21802

0.21874

16

0.22354

0.19581

0.22357

0.21662

0.21686

0.21742

17

0.22354

0.19581

0.22356

0.21806

0.21915

0.22114

18

0.22354

0.19581

0.22356

0.21802

0.21882

0.22031

19

0.22354

0.19581

0.22356

0.21745

0.21804

0.21918

20

0.22354

0.19581

0.22355

0.21643

0.21687

0.21781

21

0.22354

0.19581

0.22355

0.21790

0.21916

0.22147

22

0.22354

0.19581

0.22355

0.21788

0.21883

0.22060

23

0.22354

0.19581

0.22355

0.21733

0.21804

0.21943

24

0.22354

0.19581

0.22355

0.21633

0.21688

0.21803

25

0.22354

0.19581

0.22355

0.21781

0.21916

0.22166

Data for Figure 3 - Government Consumption

Period

Default

Commitment

0.00

0.25

0.50

0.75

1

0.06854

0.09281

0.06853

0.07005

0.07145

0.07410

2

0.06854

0.09281

0.06853

0.07078

0.07235

0.07529

3

0.06854

0.09281

0.06853

0.07133

0.07305

0.07624

4

0.06854

0.09281

0.06854

0.07173

0.07361

0.07703

5

0.06854

0.09281

0.06854

0.07012

0.07145

0.07403

6

0.06854

0.09281

0.06854

0.07086

0.07235

0.07517

7

0.06854

0.09281

0.06854

0.07141

0.07305

0.07610

8

0.06854

0.09281

0.06854

0.07180

0.07360

0.07689

9

0.06854

0.09281

0.06854

0.07016

0.07145

0.07397

10

0.06854

0.09281

0.06854

0.07090

0.07235

0.07509

11

0.06854

0.09281

0.06854

0.07145

0.07304

0.07602

12

0.06854

0.09281

0.06854

0.07184

0.07360

0.07681

13

0.06854

0.09281

0.06854

0.07017

0.07144

0.07393

14

0.06854

0.09281

0.06854

0.07093

0.07234

0.07504

15

0.06854

0.09281

0.06854

0.07147

0.07304

0.07596

16

0.06854

0.09281

0.06854

0.07187

0.07360

0.07676

17

0.06854

0.09281

0.06854

0.07018

0.07144

0.07390

18

0.06854

0.09281

0.06854

0.07094

0.07234

0.07501

19

0.06854

0.09281

0.06854

0.07148

0.07304

0.07593

20

0.06854

0.09281

0.06854

0.07188

0.07360

0.07673

21

0.06854

0.09281

0.06854

0.07019

0.07144

0.07389

22

0.06854

0.09281

0.06854

0.07094

0.07234

0.07499

23

0.06854

0.09281

0.06854

0.07149

0.07304

0.07592

24

0.06854

0.09281

0.06854

0.07188

0.07360

0.07671

25

0.06854

0.09281

0.06854

0.07019

0.07144

0.07388

Data for Figure 3 - Lambda

Period

Default

Commitment

0.00

0.25

0.50

0.75

1

0.00000

-0.53644

-0.09748

-0.08849

-0.08134

-0.06839

2

0.00000

-0.53644

-0.19488

-0.17153

-0.15719

-0.13164

3

0.00000

-0.53644

-0.29222

-0.24960

-0.22805

-0.19026

4

0.00000

-0.53644

0.00000

0.00000

0.00000

0.00000

5

0.00000

-0.53644

-0.09737

-0.08690

-0.08145

-0.07128

6

0.00000

-0.53644

-0.19470

-0.16867

-0.15738

-0.13686

7

0.00000

-0.53644

-0.29199

-0.24573

-0.22830

-0.19735

8

0.00000

-0.53644

0.00000

0.00000

0.00000

0.00000

9

0.00000

-0.53644

-0.09731

-0.08601

-0.08150

-0.07286

10

0.00000

-0.53644

-0.19460

-0.16707

-0.15748

-0.13974

11

0.00000

-0.53644

-0.29187

-0.24357

-0.22844

-0.20126

12

0.00000

-0.53644

0.00000

0.00000

0.00000

0.00000

13

0.00000

-0.53644

-0.09728

-0.08552

-0.08154

-0.07375

14

0.00000

-0.53644

-0.19455

-0.16618

-0.15753

-0.14135

15

0.00000

-0.53644

-0.29180

-0.24237

-0.22852

-0.20344

16

0.00000

-0.53644

0.00000

0.00000

0.00000

0.00000

17

0.00000

-0.53644

-0.09726

-0.08525

-0.08155

-0.07424

18

0.00000

-0.53644

-0.19452

-0.16569

-0.15757

-0.14224

19

0.00000

-0.53644

-0.29176

-0.24171

-0.22856

-0.20465

20

0.00000

-0.53644

0.00000

0.00000

0.00000

0.00000

21

0.00000

-0.53644

-0.09725

-0.08510

-0.08156

-0.07452

22

0.00000

-0.53644

-0.19450

-0.16542

-0.15758

-0.14274

23

0.00000

-0.53644

-0.29174

-0.24134

-0.22859

-0.20534

24

0.00000

-0.53644

0.00000

0.00000

0.00000

0.00000

25

0.00000

-0.53644

-0.09725

-0.08501

-0.08157

-0.07467

Data for Figure 3 - Capital

Period

Default

Commitment

0.00

0.25

0.50

0.75

1

0.87335

1.12224

0.87335

0.87335

0.87335

0.87335

2

0.87335

1.12224

0.87291

0.87624

0.88193

0.89292

3

0.87335

1.12224

0.87253

0.87306

0.88421

0.90611

4

0.87335

1.12224

0.87220

0.86504

0.88149

0.91414

5

0.87335

1.12224

0.87192

0.85312

0.87472

0.91798

6

0.87335

1.12224

0.87167

0.85881

0.88311

0.93139

7

0.87335

1.12224

0.87146

0.85801

0.88523

0.93931

8

0.87335

1.12224

0.87128

0.85205

0.88236

0.94282

9

0.87335

1.12224

0.87113

0.84190

0.87548

0.94274

10

0.87335

1.12224

0.87099

0.84915

0.88376

0.95276

11

0.87335

1.12224

0.87088

0.84968

0.88579

0.95779

12

0.87335

1.12224

0.87078

0.84486

0.88285

0.95880

13

0.87335

1.12224

0.87069

0.83570

0.87590

0.95656

14

0.87335

1.12224

0.87062

0.84381

0.88412

0.96470

15

0.87335

1.12224

0.87055

0.84508

0.88610

0.96812

16

0.87335

1.12224

0.87050

0.84088

0.88312

0.96774

17

0.87335

1.12224

0.87045

0.83227

0.87613

0.96429

18

0.87335

1.12224

0.87041

0.84086

0.88432

0.97140

19

0.87335

1.12224

0.87038

0.84254

0.88627

0.97391

20

0.87335

1.12224

0.87034

0.83869

0.88326

0.97275

21

0.87335

1.12224

0.87032

0.83038

0.87626

0.96863

22

0.87335

1.12224

0.87030

0.83924

0.88443

0.97515

23

0.87335

1.12224

0.87028

0.84114

0.88637

0.97716

24

0.87335

1.12224

0.87026

0.83748

0.88335

0.97557

25

0.87335

1.12224

0.87025

0.82934

0.87633

0.97107

Figure 4: Welfare Gains on π axis

Data for Figure 4

Probability

Welfare

1.00

0.036549

0.75

0.019559

0.50

0.010098

0.25

0.004822

0.00

0.000000

Figure 5: Welfare Gains on expected time axis

Data for Figure 5

Expected Time

Welfare

4.00

0.019559

2.00

0.010098

1.33

0.004822

1.00

0.000000

0.00

0.000000

Footnotes

* We are specially thankful to Albert
Marcet for many discussions and encouragement on this project. We
are grateful to Klaus Adam, Robert Flood, Dale Henderson, Ramon
Marimon, Francesc Ortega, Michael Reiter, Thijs van Rens, Will
Roberds and seminar participants at Federal Reserve Board, UC
Davis, Duke, University College of London, Federal Reserve of
Atlanta, Federal Reserve of San Francisco, Warwick, IIES Stockholm,
IMF, Institute for Advanced Studies in Vienna, Banco de Portugal
and Pompeu Fabra for helpful comments. Financial support
Fundação para a Ciência e Tecnologia (Nunes) is
gratefully acknowledged. Any remaining errors are our own. The
views expressed in the paper are those of the authors and do not
necessarily reflect those of the Board of Governors or the Federal
Reserve System. Return to text

1. There are several issues on the
enforcement and coordination of trigger strategies. Firstly, agents
may not be able to learn such strategies because the punishment
never occurs in equilibrium. Secondly, many atomistic private
agents would need to develop and coordinate on highly sophisticated
expectations mechanisms. Thirdly, if the punishment occurred, it is
not clear that the economy would not renegotiate and enforce a
better equilibrium. Return to
text

2. The reader is referred to Krusell et al. (2006) and Debortoli and Nunes (2007). Return to
text

3. As discussed in Martin (2007), it is important that at least some
depreciation is not tax deductible, as assumed for instance in
Greenwood et al. (1998). If this is not the case, the
current capital tax is not viewed by the current government as
distortionary and an equilibrium in such economy may not exist. In
some developed economies, there is a tax allowance for accounting
depreciation, which differs from the actual depreciation. If there
is excess depreciation due to a high capacity utilization such
depreciation would still not be tax deductible. Return to text

4. The class of models that our
methodology is able to handle is fairly general and has the same
requirements of Marcet and Marimon (1998). The
separability in Eq. (12) is not
necessary and the terms
can all interact
in a multiplicative way. Our methodology is also able to handle
participation constraints and other infinite horizon constraints,
as also described in Marcet and Marimon (1998). Return to
text

5. For further discussions on this issue
see Klein et al. (2007). Return to
text

6. only contains the history
and
similarly the set
only contains the history
. Return to text

7. Debortoli and Nunes (2007),
relax this assumption and focus on political disagreement
issues. Return to text

9. As it is common in the time-consistent
literature and also in the optimal taxation literature we do not
prove that the optimal policy function is unique. Nevertheless, we
found no evidence of multiple solutions. Return to text

10. Note that, when default occurs, the
lagrange multiplier is set to zero and cannot be used to influence
incoming planners. Return to
text

12. The zero long run tax on capital
still holds for a variety of cases (including balanced budget) as
shown by Judd (1985). Return to text

13. We also considered initializing
capital at other steady states or expressing in consumption units of the alternative regime. This
means computing as
. The results remain unchanged. Return to text

14. In Barro-Gordon models the welfare
loss penalizes quadratically deviations of inflation from zero and
deviations of the output gap from a target level. The inflation and
output gap under commitment are nearly zero. Under discretion the
inflation is quite high and the output gap is still zero. Since
standard calibrations give a much higher weight to inflation
deviations in the loss function, the gains from commitment are
substantial. Return to text

15. If the probability function depended
on a non-predetermined variable, then the FOCs would be different
when a planner starts and when it is already in power. This would
introduce another source of time inconsistency. Return to text

16. More formally, one could model a
continuum of agents on a real interval between 0 and 1. All agents
would be equal, and therefore their decisions would be equivalent
to a representative agent, who takes aggregate capital as
given. Return to text

17. For the purpose of this proof one has
to include as a state variable. This is
only convenient for this proof and as discussed later is not
necessary for the numerical work. Return
to text

18. The symbol
denotes the partial derivative
of the function with respect to . We suppressed the arguments of the functions for
readability purposes. Return to
text

19. Note that in performing such
operation the term
multiplies
. If there is a default that
term disappears. If there is no default it is still the case that
. Therefore, the
original FOC and the changed FOC in Eq. (50,
51) are
equivalent. Return to text